i don't think this is a correct solution. you might need to look at it more carefully.

I assume you are right that I made a mistake somewhere! I'm an applied guy so I'm not very good at this pure stuff. But I don't see where I made my error (well, to be honest, I didn't look very hard). Could you drop me a hint?

[ ] Let be a finite dimensional vector space over some field and Let be a finite set of the subspaces of and suppose that for any with we have

Prove that [tex]\bigcap_{W \in \mathcal{A}

[From now on I'll rate the problems that I give in here. It starts with one star, which means "fairly easy", and goes up to 5 stars, which means "unfairly difficult"! haha]

An efficient but not very clean solution; there must be a way to make it nicer.

Spoiler:

Assume . We have to prove the existence of a subset of of cardinality at most such that .

For every , choose a linear map such that (some projection on a complement subspace, for instance). Finally, let us introduce the linear map given by . Its kernel is due to the assumption, hence is injective, and its image has dimension . Consider a matrix of , consisting in piling up the matrices of ( ); then we can find independent rows. Let be the subspaces corresponding to these rows ( if two rows come from the same "submatrix"). The map has rank by the previous choice, hence its kernel (i.e. ) is . This concludes.